(I)
On the simple end of the spectrum, we have theories where
the DM may be the only accessible state to our experiments. In such a
case, effective field theory (EFT) allows us to describe the DM-SM
interactions mediated by all kinematically inaccessible particles in
a universal way. The DM-EFT approach [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}
has proven to be very useful in the analysis of LHC Run I data, because
it allows to derive stringent bounds on the “new-physics" scale
\(\Lambda \)  that suppresses the higher-dimensional operators. Since for
each operator a single parameter encodes the information on all the
heavy states of the dark sector, comparing LHC bounds to the limits
following from direct and indirect DM searches is straightforward in
the context of DM-EFTs.
(II)
The large energies accessible at the LHC call into question
the momentum expansion underlying the EFT approximation [4]}, [8]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]},
and we can expand our level of detail toward simplified DM models (for
early proposals see for example [18]}, [19]}, [20]}, [21]}, [22]}, [23]}).
Such models are characterized by the most important state mediating the
DM particle interactions with the SM, as well as the DM particle itself.
Unlike the
DM-EFTs, simplified models are able to describe correctly the full
kinematics of DM production at the LHC, because they resolve the EFT
contact interactions into single-particle \(s\) -channel or \(t\) -channel
exchanges. This comes with the price that they typically involve not
just one, but a handful of parameters that characterize the dark sector and
its coupling to the visible sector.
(III)
While simplified models capture some set of signals
accurately at LHC energies (and beyond), they are likely to miss
important correlations between observables. Complete DM models close
this gap by adding more particles to the SM, most of which are not
suitable DM candidates. The classical example is the Minimal
Supersymmetric SM (MSSM), in which each SM particle gets its own
superpartner and the DM candidate, the neutralino, is a weakly
interacting massive particle. Reasonable phenomenological models
in this class have of order 20 parameters, leading to varied visions of
DM.
At the same time, they build-in correlations from symmetry-enforcing
relations among couplings, that would look like random accidents in a
simplified model description.
Complete DM models can in principle answer any question satisfactorily,
but one might worry that their structure is so rich that it is impossible
to determine unambiguously
the underlying new dynamics from a finite amount of data
(“inverse problem") [24]}.

